Let $f(x)=\frac{1}{4} X+4$, $g(x)=3 X^2$, And \$h(x)=4 X+12$[/tex\].Perform The Indicated Composition And State The Domain.15. $f \circ H$16. $h \circ G$17. $f \circ F$
Introduction
In mathematics, composition of functions is a fundamental concept that allows us to combine multiple functions to create new functions. Given two functions, f(x) and g(x), the composition of f and g, denoted as (f ∘ g)(x), is defined as f(g(x)). In this article, we will explore the composition of three functions: f(x) = (1/4)x + 4, g(x) = 3x^2, and h(x) = 4x + 12. We will perform the indicated composition and determine the domain of each resulting function.
Composition of Functions
Composition of f and h
To find the composition of f and h, we need to substitute h(x) into f(x) in place of x. This means we will replace x in f(x) = (1/4)x + 4 with h(x) = 4x + 12.
(f ∘ h)(x) = f(h(x)) = f(4x + 12)
Now, we substitute h(x) into f(x):
(f ∘ h)(x) = (1/4)(4x + 12) + 4
To simplify the expression, we can distribute the 1/4 to both terms inside the parentheses:
(f ∘ h)(x) = x + 3 + 4
Combine like terms:
(f ∘ h)(x) = x + 7
Therefore, the composition of f and h is (f ∘ h)(x) = x + 7.
Composition of h and g
To find the composition of h and g, we need to substitute g(x) into h(x) in place of x. This means we will replace x in h(x) = 4x + 12 with g(x) = 3x^2.
(h ∘ g)(x) = h(g(x)) = h(3x^2)
Now, we substitute g(x) into h(x):
(h ∘ g)(x) = 4(3x^2) + 12
To simplify the expression, we can distribute the 4 to both terms inside the parentheses:
(h ∘ g)(x) = 12x^2 + 12
Therefore, the composition of h and g is (h ∘ g)(x) = 12x^2 + 12.
Composition of f and f
To find the composition of f and f, we need to substitute f(x) into f(x) in place of x. This means we will replace x in f(x) = (1/4)x + 4 with f(x) = (1/4)x + 4.
(f ∘ f)(x) = f(f(x)) = f((1/4)x + 4)
Now, we substitute f(x) into f(x):
(f ∘ f)(x) = (1/4)((1/4)x + 4) + 4
To simplify the expression, we can distribute the 1/4 to both terms inside the parentheses:
(f ∘ f)(x) = (1/16)x + 1 + 4
Combine like terms:
(f ∘ f)(x) = (1/16)x + 5
Therefore, the composition of f and f is (f ∘ f)(x) = (1/16)x + 5.
Domain of the Composed Functions
To determine the domain of each composed function, we need to consider the restrictions imposed by each individual function.
- For (f ∘ h)(x) = x + 7, the domain is all real numbers, since there are no restrictions on x.
- For (h ∘ g)(x) = 12x^2 + 12, the domain is all real numbers, since there are no restrictions on x.
- For (f ∘ f)(x) = (1/16)x + 5, the domain is all real numbers, since there are no restrictions on x.
In conclusion, the composition of functions is a powerful tool in mathematics that allows us to create new functions by combining existing ones. By understanding the composition of functions, we can solve a wide range of mathematical problems and explore new mathematical concepts.
Conclusion
In this article, we explored the composition of three functions: f(x) = (1/4)x + 4, g(x) = 3x^2, and h(x) = 4x + 12. We performed the indicated composition and determined the domain of each resulting function. The composition of functions is a fundamental concept in mathematics that has numerous applications in various fields, including science, engineering, and economics. By mastering the composition of functions, we can solve complex mathematical problems and explore new mathematical concepts.
References
- [1] "Composition of Functions." Khan Academy, Khan Academy, www.khanacademy.org/math/algebra/x2f6f6c6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d6d
Composition of Functions: A Q&A Guide =====================================
Introduction
In our previous article, we explored the composition of functions and performed the indicated composition of three functions: f(x) = (1/4)x + 4, g(x) = 3x^2, and h(x) = 4x + 12. In this article, we will answer some frequently asked questions about composition of functions.
Q&A
Q: What is the composition of functions?
A: The composition of functions is a way of combining two or more functions to create a new function. Given two functions, f(x) and g(x), the composition of f and g, denoted as (f ∘ g)(x), is defined as f(g(x)).
Q: How do I find the composition of two functions?
A: To find the composition of two functions, you need to substitute one function into the other in place of x. For example, to find the composition of f and g, you would substitute g(x) into f(x) in place of x.
Q: What is the domain of a composed function?
A: The domain of a composed function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of x for which the function is valid.
Q: Can I compose more than two functions?
A: Yes, you can compose more than two functions. For example, you can find the composition of f, g, and h by substituting h(x) into f(g(x)).
Q: What are some real-world applications of composition of functions?
A: Composition of functions has numerous real-world applications in various fields, including science, engineering, and economics. For example, in physics, the composition of functions is used to model the motion of objects under the influence of gravity. In engineering, the composition of functions is used to design and optimize complex systems.
Q: How do I determine the domain of a composed function?
A: To determine the domain of a composed function, you need to consider the restrictions imposed by each individual function. For example, if one function has a domain of all real numbers, but another function has a domain of only positive real numbers, then the composed function will have a domain of only positive real numbers.
Q: Can I use composition of functions to solve equations?
A: Yes, you can use composition of functions to solve equations. For example, if you have an equation of the form f(x) = g(x), you can use composition of functions to find the solution.
Q: What are some common mistakes to avoid when working with composition of functions?
A: Some common mistakes to avoid when working with composition of functions include:
- Not substituting the correct function into the other function
- Not considering the domain of each individual function
- Not simplifying the expression correctly
Conclusion
In this article, we answered some frequently asked questions about composition of functions. We hope that this article has provided you with a better understanding of composition of functions and how to apply it in various situations.